An uncountable countable set
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From: Ross A. Finlayson on 25 Oct 2006 22:18 Dik T. Winter wrote: > In article <1161825225.141821.66550(a)e3g2000cwe.googlegroups.com> "Ross A. Finlayson" <raf(a)tiki-lounge.com> writes: > ... > > Cardinals are ordinals and in a set theory: sets. > > > > Generally it's known that you get cardinals from the previous using the > > powerset operation, > > Through what powerset operation do you get the cardinal 3? > > > and ordinals from the previous using the successor > > operation. > > Through what successor operation do you get the ordinal omega? > > > So, consider the formation of the ordinals where the powerset of the > > previous ordinal is the successor. > > In that way you do not construct the ordinals, but the Finlayson numbers. > They go like 0, 1, 2, 4, 8, 16, 32, ... > > > You might say "the von Neumann ordinals are simpler than that" and also > > they contain less information. > > Well, at least they contain ordinal numbers like 3, 5, 6, 7 etc. > -- > dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 > home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ I understand it. Please identify something you see as incorrect or don't understand. There's no universe in ZF. So, quantify over sets. Basically, I plan to cut and paste the posts into other documents, in the evolving discussion. Ross Hi Dik, You ask where you get 3, eg from 2, using the powerset. This is where 2 was the powerset of 1 and 1 the powerset of 0. They're ordinals. About limit ordinals like omega, N E N, that's about how the successor of any finite number is finite, and concidentally how the successor of every finite number is not. The ordinals as powersets have some more structure that can be used in the explanation of algorithms upon those mechanistic ordinals as numbers with the corresponding properties of what you call the natural numbers. It is only in the last few years that articles in the BSL/JSL admit the reals a natural well-ordering. So, did you find that counterexample? Ross
From: David Marcus on 26 Oct 2006 00:06 Ross A. Finlayson wrote: > MoeBlee wrote: > > > There are a few correct statements in there, but most of the above is > > just plainly incorrectl mixed with gibberish. > > > > I'm just cursious to know what satisfcation you derive from typing up > > and posting such nonsense, post after post after post, year after year > > after year. > > > > MoeBlee > > I understand it. > > Please identify something you see as incorrect or don't understand. The next thing you say is a good example of something unintelligible: > There's no universe in ZF. So, quantify over sets. What could that possibly mean? It doesn't appear to mean anything. -- David Marcus
From: Ross A. Finlayson on 26 Oct 2006 00:39 David Marcus wrote: > Ross A. Finlayson wrote: > > MoeBlee wrote: > > > > > There are a few correct statements in there, but most of the above is > > > just plainly incorrectl mixed with gibberish. > > > > > > I'm just cursious to know what satisfcation you derive from typing up > > > and posting such nonsense, post after post after post, year after year > > > after year. > > > > > > MoeBlee > > > > I understand it. > > > > Please identify something you see as incorrect or don't understand. > > The next thing you say is a good example of something unintelligible: > > > There's no universe in ZF. So, quantify over sets. > > What could that possibly mean? It doesn't appear to mean anything. > > -- > David Marcus Fascinating. Where do the sets come from? Ross
From: RLG on 26 Oct 2006 03:06 "Tony Orlow" <tony(a)lightlink.com> wrote in message news:453fac14(a)news2.lightlink.com... > RLG wrote: >> "Tony Orlow" <tony(a)lightlink.com> wrote in message >> news:453e4a3f(a)news2.lightlink.com... >> >> From my reading of this issue the vase is empty at noon, as David Marcus >> says. But Tony, I have a question for you. Suppose we put one more ball >> into the vase, at any time before noon, and that ball is labeled "oo". >> At exactly noon that ball is removed from the vase. At noon is the vase >> empty or does it contain the ball labeled "oo"? >> >> -R >> > > Hi RLG. Welcome to the conversation. > > According to the experiment, all balls inserted and removed are finite, so > that doesn't really apply. Every ball n is inserted at time -1/n > (or -1/2^n originally, but it doesn't matter), so ball oo cannot be > inserted before noon. > > But, if you want to entertain the idea of inserting an extra ball named > "oo", or "Bill", or "RLG", the addition of a ball is not going to make the > vase any more empty. I wonder what logical implication you think that > has.... My apologies if the logical implication was unclear. What I was trying to get at was the issue of how well point set theory does or does not apply to the continuum. Suppose a ball labeled "oo" is put into the vase five minutes before noon. I asked that if at exactly noon the ball labeled "oo" is removed from the vase is the vase empty at noon or does it contain the ball labeled "oo"? Is the vase time for the "oo" ball [-5,0] or [-5,0) and is there a difference between these two in terms of temporal duration? In the [-5,0] case the ball is in the vase at the instant t=0 but not at any time t>0. Since the ball is in the vase at t=0 it is not removed at that time so it must be removed at some time t>0. But for every t>0 there is a (t/2)>0 such that t>(t/2)>0 and so at no t>0 is the ball removed from the vase. It seems that in the [-5,0] case there is no instant at which the "oo" ball is removed from the vase. In the [-5,0) case the ball is not in the vase at time t=0 so it seems it should have been removed at some time t<0. But the "oo" ball is in the vase for all t<0 and so at no t<0 is the "oo" ball removed from the vase. It seems that in the [-5,0) case there is no instant at which the "oo" ball is removed from the vase. So I ask you again, if the "oo" ball is removed from the vase at exactly noon, is it in the vase at exactly noon or not? Maybe this is a joke question. Before considering that question too deeply, consider another question. The length of the two intervals [0,1] and [0,1) is both 1. In point set theory these are two different sets, the latter set is identical to the former set except it does not contain the number 1. Yet, in terms of distance, these distinctions make no sense. If you have a stick one meter long and take nothing at all off its right end, you haven't changed anything. In terms of distance and time distinctions like [-5,0], [-5,0), [0,1], and [0,1) are meaningless. This raises the question as to how well point set theory applies to the continuum and I remember reading somewhere that Godel had concerns about this. In my opinion, point set theory is the best tool presently available for studying the continuum but I recognize that there are some limitations to it. This whole issue that you and other have been arguing is called the Ross-Littlewood paradox. I suggest you read about it at http://en.wikipedia.org/wiki/Supertask. Wikipedia phrases it this way: "Suppose you had a jar capable of containing infinitely many marbles, and an infinite collection of marbles labeled 1, 2, 3, and so on. At t=0, marbles 1 to 10 are placed in the jar, at t=1/2 11 to 20 are placed in the jar but marble 1 is taken out. At t=3/4 marbles 21 to 30 are put in the jar and marble 2 is taken out: in general at time t=1-(1/2)^n, the marbles (10*n + 1) to (10*n + 10) are placed in the jar and marble n is taken out. The question is: How many marbles are in the jar at t=1?" Here is my take on the whole Ross-Littlewood issue. If all the marbles are labeled as described above, they should all be out of the jar at the end of the supertask. So I agree with your antagonists on this point. But if they were labeled differently that need not be the case. Suppose at t=0 ten marbles are placed in the jar but they are labeled (1,2,3,4,5,w,w+1,w+2,w+4,w+5) and at t=1/2 ten more marbles are placed in the jar labeled (6,7,8,9,10,w+6,w+7,w+8,w+9,w+10) etc and each marble labeled with the nth natural number is removed at time t=1-(1/2)^n. At the end of the supertask all marbles labeled with natural numbers will be out of the jar and all the marbles labeled with transfinite ordinal numbers will be left in the jar. In that case there is an infinite number of marbles in the jar at the end of the supertask but not one of them is labeled with a natural number. Some related links: http://en.wikipedia.org/wiki/Supertask http://plato.stanford.edu/entries/spacetime-supertasks/ http://en.wikipedia.org/wiki/Category:Supertasks -R
From: Han de Bruijn on 26 Oct 2006 03:43
Tony Orlow wrote: > David Marcus wrote: > >> cbrown(a)cbrownsystems.com wrote: >> >>> stephen(a)nomail.com wrote: >>> >>>> With the added surreal twist that the limit of the number >>>> of unnumbered balls in the vase as we approach noon is 0, >>>> but the number of unnumbered balls in the vase at noon is >>>> infinite. :) >>> >>> I think his response, when I pointed this out to him, was either "Oh, >>> shut up!" or "Whatever." >> >> That is consistent with my suggestion that Tony is reasoning by >> imagining a vase filling up. If you visualize the vase filling up in >> your mind, you don't see the unnumbered balls in the picture. > > If you have an infinite ocean wit 10 liter/sec flowing in, and 1 > liter/sec flowing out (and no evaporation), will it ever empty? No. Same > difference. Ah! With Tony's ocean, that nice picture of mine really has become applicable: http://hdebruijn.soo.dto.tudelft.nl/jaar2006/ballen.jpg Han de Bruijn |